Mathematics

\(\cos(\frac {\pi} {4})=\frac {1} {\sqrt{2}}\) \(\cos(\frac {\pi} {5})=\frac {1+\sqrt{5}} {4}\) \(\cos(\frac {\pi} {6})=\frac {\sqrt{3}} {2}\) Is cosine of any fraction of π an algebraic number?

This is a short guide to using the new LaTeX system that has been implemented on the boards. First off, for those who don't know what LaTeX is, a short description. LaTeX is, to all intents and purposes, a fully fledged math typesetting system - basically put, you can write math with it. It's a very flexible and hence very advanced piece of software, and the syntax for it is quite complex, but fairly easy to learn for typesetting smaller equations. On scienceforums.net, we've implemented a small LaTeX system to allow you to typeset equations (in other words, cut out all the x^2 stuff and make things easier to read for everyone). The basic principle behind it is this: you…

For any prime number n > 3, n mod 6 = 1 or 5. any prime number n > 3, n mod 3 = 1 or 2. The same prime numbers are in column 1 either way and the primes from column 2 (mod 3) are in column 5 (mod 6). Are there 2 kinds of prime numbers? Is there a name for these primes? The attached png is the primes mod 2,3,6, and 7.

* = multiplication i = inverse (of) multiplication 2 * 2 = 4 4 i 2 = 2 The reason I post this is because I don't see it appearing as part of arithmetic operations elsewhere apart from some references to inverse (e.g. link: https://en.wikipedia.org/wiki/Multiplicative_inverse) but probably not exactly what I am talking about.

As far I can discern when inventing combinatorics I also found the product rule for differentiation and derivatives. Not "Isaac Newton" or some other name that I can't confirm ever existed. When pertaining to the scientific method, empirical evidence is based on observation which I don't rely on as heavily as mathematics for separating fact from fiction. I know I don't get anything out of sharing an arithmetic that can literally be used to earn a person money for cracking a password combination or solving any type of Rubiks cube. I should be retired for this alone. I only share it because I know some things that are potentially worth a lot more. Just realize wheneve…

I have a question related to this topic. If we create a three dimensional matrix, and give each element of the matrix a name that would represent another three dimensional matrix, similar to how computers do it, wouldn't it be possible to visualize n-dimensional matrices? Inside each element of this matrix there would be other 3D elements. This makes visualizing, in this case, six dimensional matrices possible, no?

Hi I give here the Geometric Model and representation of Walker's Equation and proposal of an Equation od Infinite Series which I call 'Walker's Series' which appears to have escaped many ! Proof of Walker's equation.docx - Google D Dear Moderator, I tried to remove the duplicate images in this post but couldn't succeed Plz remove the repeating pics TY !

In short, I am curious about how this even works and I want to know because I enjoy math, so I decided to ask people with a higher education than mine and who definitely know more about math than I do. So my understanding of graphing in three dimensions to give you an idea of my level of understanding of the concept so you can explain it better is essentially this: In order to graph in more than two dimensions we must simply add a third axis: Z. Z represents the depth axis and intersects X and Y at the origin vertically relative to the plane that X and Y form. The Z axis forms a plane with the Y axis vertically and the X axis horizontally. Graphing…

(x^3/N) approximately= (x^2/y) (25/17) approximately= (85/58) Thoughts?

I have a paper that, IMHO, proves the Collatz conjecture. I will be looking at where to get it confirmed. I welcome comments. Abstract: The Collatz tree can be formed naturally, resulting in a good looking tree, but the numbers seem chaotic. By reforming the tree, the tree becomes weird, but the numbers look nice. And it becomes clear that the hailstone algorithm and its inverse can be used to traverse down and up the tree, from the root to all integers and beyond, and back to the root (the number 1). The link is http://dbarc.net/yr2024/collatz1.0.pdf

Can anyone explain in one paragraph or equation how the Riemann hypothesis implies a pattern in Prime numbers? The hypothesis an easy enough to follow the problem, but the pattern in Primes is not clear to me. I’m sure a Google search may be helpful, but I wanted to work it through on my own. I find it helpful to take one small part if the problem and see what it does. Take the book Practical Cryptography. It explains all the ciphers. They are all open source. Doesn’t mean you can solve them but it lets you see the inner workings. That is what I need an explanation of the patterns in Primes in the Riemann hypothesis. I hope you understand why I just don’t g…

I recently came across this equation on social media: At first I thought it would the answer after simplifying the equation would be and thus 12 + 6 = 18, but inputting into my calculator gets me 8 instead. Inputting the reverse () gives me my expected answer of 18. Is it an inherent property of percentages to take what's to the left?

Greetings to all. How to calculate the differential in hardware needs (a radiator size and coolant mass and ...) ? A plain simple automobile uses a number of litres of water as coolant on a radiator of such and such dimensions. If the water specific heat capacity is 4,181 J/kg°C , a very high number compared to many other fluids- is replaced by plain motor oil / transmission fluid; what does it translate to ? --> Doubling the area/volume/fanning of its radiator ? Tripling ? As replacement of water based coolant in ICE engines, transmission fluid can: - Eliminate rust progress or prevent it, - Eliminate boiling at operating temperatures, - Eliminate …

I have watched a video about the p-adic numbers: Then, there is the Ramanujan's summation which is used in quantum mechanics (Casimir effect): 1+2+3+4+5+6...=-1/12 https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension This forum does not support Latex maths? Then, we have one more such summation: 1+x+x*x+x*x*x+x*x*x*x+x*x*x*x*x...=-1/(1-x) This is simple for x<1 and for x>1 we get: 1+2+4+8+16+32+64...=-1 1+10+100+1000+10000...=-1/9 1+20+400+8000+160000...=-1/19 Can we obtain all these sums with the p-adic numbers?

I came across an intriguing iterative algorithm for solving a nonlinear equation of the form ln(f(x))=0 , which differs from the classical Newton's method. This method utilizes a logarithmic difference to calculate the next approximation of the root. A notable feature of this method is its faster convergence compared to the traditional Newton’s method. The formula for the method is as follows: $$x_{n+1} = \frac{\ln(f(x + dx)) - \ln(f(x))}{\ln(f(x + dx)) - \ln(f(x)) \cdot \frac{x_n}{x + dx}} \cdot x_n$$ Example: Using the classical Newton's method, the initial approximation x0=111.625 leads to x1=148.474 U…

As per the link overhead, the jackpot increases by $2E6 for every ball removed, I know. But where did this(69−2B) stem from? I feel like the author, Paul Sinclair, pulled it out of a hat! The maximum Gold Ball Draw jackpot is $68E6, not $69E6. Then why did Paul Sinclair write 69?

1. Disregarding the messier LHS, how can I intuit the simpler RHS? I know [math]\color{green}{\dfrac{1E6}{T} = \Pr(}[/math] winning $1E6 in the Gold Ball Draw). 2. But why multiply this [math]\color{green}{\dfrac{1E6}T}[/math] by [math]\color{red}{\dfrac{69 - B}B}[/math]? 3. What does [math]\color{red}{\dfrac{69 - B}{B}}[/math] mean?

What’s wrong with my reasoning in purple? Please pinpoint which sentence and step fails. I seek intuition. I DON’T want proofs, or formal arguments. Thanks! Presuppose lotteries let players pick any integer N≥32. Picking unpopular integers lowers your probability of winning lotteries. Why? Winning numbers range randomly from 1 to N. But you artificially narrow yourself to [32,N]. By disregarding [1,31], you flout the lottery’s random distribution of winning integers! As [1,N] contains more integers than [32,N], picking numbers ∈ [1,N] proffers more chances to win than picking ∈ [32,N]. Q.E.D.

what would it be then?

Hello all. Am on first steps to make a solar thermal water pump for drip irrigation purposes. Assume 60% efficiency of solar thermal irradiation (600W/m2. Assume average daylight temperature 25C. Assume average night temperature 15C. Or, make those parameters real, entered in the formula. Area of sun exposure as parameter, say =1 m2 (Tank lid) Capacity of reservoir as parameter, say = 1 m3 Volume of air, =Va. Volume of water =Vw. For a desired pumping height, what would be the flow in 24 hours ? [Plus whatever am missing] In the hermetic reservoir, the solar heat expands the air pushing the water up during the day, and suctioning water …

3x+1 if odd x/2 if even So why not create a sieve of multiples of 2, 4, and 8? When x equals a logarithm of 2, 4, or 8 then x/2 repeated equals 2. 2*2*2*2… 4*4*4*4… 8*8*8*8… These 3 logarithms of 2, 4, and 8 encompass all the numbers that reduce to 2 when repeatedly divided by 2. What am I missing? Did I break the rules of the pattern? I am trying real hard not to factor semiPrimes. I did not research the problem yet. I am only working with 3x+1. I realize that this is more complex with larger numbers. I am just stating a simple example.

I have been researching Quantum Algebra for a few days. I have a few questions about it in hopes that people can help me on here. 1. How do quantum groups differ from classical groups in mathematics? 2. Can you explain the concept of Hopf algebras and their significance in quantum algebra?

6 balls are drawn from a total of 49 balls randomly. Order does not matter, as long as 6 balls are matched, first prize is won, what is the probability of winning the first prize? Another question is horse racing. For certain three matches, which may have 10 to 14 horses in total, different horses may have different conditions and abilities so the winning odds of each horse may not be as simple as 1/10 or 1/14. You choose three horses from each of the three matches so you are choosing 9 horses in total, if all 9 horses in three matches finish for the 1st, 2nd and 3rd place, order isn't counted, i.e. ABC, ACB, BAC, BCA, CAB, CBA don't matter.), then you win the jackpo…

Sales tax Pyramid Scheme Ok say I buy something for$100 + $6.00 sales tax. Then the person I bought from buys something for $100 + $6.00 sales tax. And so on and so forth for n, the number of money transactions. Does that mean that on $100 the government just make $6.00 * n. And after $100 / $6 times the government is making more money than the original $100? So the act of saying $100 is worth so much in gold, actually makes more than the value of the gold? And then transfer of money is promoted because it generates more value? And you thought bitcoin was a scam.

In a mechanical memory system, such as a tape recorder or gramophone, the input movements of sound waves are constrained by the mechanism to motions back and forth along a single line. A separate mechanism(s) adds motion along a different line to stretch the needle marks far enough that they don't overlap at frequencies within the design range. The question is, how can the speed and direction of the added movement(s) be calculated from the recording needle track? With a needle bouncing perpendicular to the length of a recording tape, if the tape moves in a straight line past the moving needle there are only two directions to choose from. Can we tell if the tape …